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diff --git a/final/main.tex b/final/main.tex index dfa5914..18d0a30 100644 --- a/final/main.tex +++ b/final/main.tex @@ -322,22 +322,22 @@ Lemma. = \big(F^{-1}(1-p_1), \dots, F^{-1}(1-p_m)\big).$$ \end{lemma} -\section{$m$-to-1 Bidders Reductions} +\section{$n$-to-1 Bidders Reductions} The goal of this section is to compare the two ways of constraining the single buyer problem that we discussed: namely $p$-exclusivity and ex-ante allocation constraints and draw a parallel in how these two notions are used to construct -$m$-to-1 bidders reductions in \citep{yao} and \citep{alaei} respectively. +$n$-to-1 bidders reductions in \citep{yao} and \citep{alaei} respectively. The notion of $p$-exclusivity introduced by \citep{yao} was crucial in his -reduction from the $k$-item $m$-buyer setting to the $k$-item single buyer +reduction from the $k$-item $n$-buyer setting to the $m$-item single buyer setting. He describes a mechanism known as \emph{Best-Guess Reduction}, which -conducts $m$ single-buyer $k$-item auctions, using an IR-IC $p$-exclusive +conducts $n$ single-buyer $m$-item auctions, using an IR-IC $p$-exclusive mechanism, for a particular value of $p$ drawn from the joint bid distribution over all buyers conditioned on the bids of all other buyers, and then combines this with the Vickrey second-price auction, showing that this mechanism has -revenue that is a constant approximation to the optimal $k$-item, $m$-buyer +revenue that is a constant approximation to the optimal $m$-item, $n$-buyer mechanism. He then defines another mechanism, \emph{Second-Price Bundling}, which is meant to heuristically approximate this combined mechanism, and shows that its revenue is also a constant approximation to the optimal mechanism. |